Taylor Series HW

# Taylor Series HW - Taylor Series HW 1 x = 0...

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Taylor Series HW 1) Construct the fourth order Taylor polynomial at 0 x = for the function. (a) 2 ( ) 1 f x x = + (b) 2 ( ) x f x e = 2) Construct the fifth order Taylor polynomial and the Taylor series for the function at 0 x = (a) 1 ( ) 2 f x x = + (b) 1 ( ) x f x e = For #3 4, use the table of common Maclaurin series from your notes. Construct the first three nonzero terms and the general term of the Maclaurin series generated by the function and give the interval of convergence. 3) sin 2 x 4) 1 2 tan x 5) Find the Taylor series generated by the function 1 ( ) 1 f x x = + at 2 x = . For #5 6, find the Taylor polynomial of order 3 generated by ( ) f x at 1 x = . 6) 3 ( ) 2 4 f x x x = + 7) 4 ( ) f x x = For #8 9, find the Taylor polynomials of orders 0, 1, 2, and 3 generated by ( ) f x at x a = . 8) ( ) sin , 4 f x x a π = = 9) ( ) , 4 f x x a = = 10) Let ( ) f x be a function that has derivatives of all orders for all real numbers. Assume (1) 4, (1) 1, (1) 3, (1) 2 f f f and f ′′ ′′′ = = − = = .

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